How do I round the edges of a side?
You can use "Bevel" on selected edges. 

Select the edges that you want to round, then hit Ctrl+B to bevel them. Scrolling your middle mouse button wheel you can choose the bevel resolution. 



Bevel comes in different "modes" that you can cycle using M during the tool's operation. Unfortunately neither of these modes can be used to force a predominant axis. For fine adjustments you will need to edit your geometry before or after applying bevel.